# Twomirror Telescopes

We introduced the topic of two-mirror telescopes in Chapter 2 with schematic diagrams of two types, Cassegrain and Gregorian, in Fig. 2.7, as well as a set of definitions of normalized parameters with which to describe any two-mirror telescope. Selected items from Section 2.5 and Table 2.1 are summarized in Table 6.3 for convenient reference.

It is instructive to study the relations between the normalized parameters in Table 6.3 because they define the bounds on the parameters for each of the possible telescope types. For all types we require that the final image is real.

If the primary is concave, hence /j positive, the requirement of a real final image means mk > 0. If m and k are positive the telescope type is Cassegrain; if k = y-ily\ = ratio of ray heights at mirror margins p = R2/Ri = ratio of mirror radii of curvature m = —s'2/s2 = ///i =transverse magnification of secondary fiP = Drj — back focal distance, or distance from vertex of primary mirror to final focal point /} and t], back focal distance in units offx and D, respectively Fi = |/j \/D — primary mirror focal ratio

W = (1 — k)fx = distance from secondary to primary mirror = location of telescope entrance pupil relative to the secondary when the primary mirror is the aperture stop mkfx = distance from secondary to focal surface F = \f\/D = system focal ratio, where / is telescope focal length

Table 6.3

Normalized Parameters for Two-Mirror Telescopes mk k_l+ß

their signs are negative the telescope type is Gregorian. In both cases \k\ < 1 to ensure that some light reaches the primary.

If the primary is convex, hence fx negative, then a real final image requires that mk is negative. In this case the secondary must be larger than the primary, hence k > 1, and m is negative. This type of telescope, with its concave secondary, is the so-called inverse Cassegrain.

The different combinations of m, k, and p are summarized in Table 6.4. It is worth noting here that among the Cassegrains with concave secondary and the inverse Cassegrains are the so-called Couder and Schwarzschild designs that will be discussed later in this chapter. The Cassegrain with flat secondary is not included in the analysis and discussion to follow.

We now proceed to find the aberration relations for two-mirror telescopes using Eqs. (5.6.11) and the aberration coefficients of the primary and secondary in Eqs. (5.6.9) and (5.6.10). Before writing the system aberration coefficients, W is written in terms of the normalized parameters: W/R2 = (k — l)/2p, and W/s2 = (k — \)/k. With these substitutions, and after straightforward but tedious algebra, the two-mirror aberration coefficients given in Table 6.5 are found. Note that these coefficients apply to any pair of conic mirrors, including pairs for which the spherical aberration is not zero. It is worth noting that Bis is the only aberration coefficient affected by the conic constant of the primary mirror. An error in Kx, such as for the Hubble Space Telescope, has no effect on the off-axis aberrations.

We can also use the condition for zero spherical aberration and rewrite the aberration coefficients in terms of Kx. Setting Bis in Table 6.5 equal to zero we find, after more algebra, the results given in Table 6.6. These results are based on a choice of locating the aperture stop at the primary mirror. When spherical aberration is zero, coma is independent of the stop location; when both SA and coma are zero, astigmatism is independent of the stop position. We will comment further on these conditions when discussing specific types of telescopes.

Table 6.4

Parameter Combinations for Two-Mirror Telescopes'

Type

Secondary

<0 <0 <0 Gregorian concave concave convex flat

<0 >1 >0 Inverse Cassegrain concave

Table 6.5

General Aberration Coefficients for Two-Mirror Telescopes